c = rand(3,1);
C = expm(hat(c));
w =[1 0 0 ]';

t = 1e-3;
C_left =  expm(hat(w*t)) * C;
%%
(C_left - C)/ t - hat(w)*C
pause()
%%
hat(w) - ((C_left - C)/t)*C'
pause()
%%
fun = @(a)C^a;
J = integral(fun, 0, 1, 'ArrayValued', true);

c_left = vee(logm(C_left));
w - J * (c_left - c)/t